First thing is:
Make sure (with a loaded rifle handy) that the beast is in fact dead, not just waiting to kick the daylights out of you. :shocking:
The heart I always save unless shot to pieces (happens a fair amount of the time).
The liver I also always save. I love to make a spread or with liver and onions.
Otherwise I quickly but carefully open the organ cavity, remove the heart and liver and Ziploc bag, and then trim the entrails away from the rib cage minding the inside loin straps and then the bladder (carefully) and the glands in one "clump". I try to leave enough of the larynx in so I have something to grab ahold of later. I take a zip/wire bone saw, push it under the pelvic bone and saw through the bone and then easily spread the bone apart and everything slides right outside the carcass in one big pile. Then trim off the glands with the gut pile.
Neat and complete.
I never have saved the glands.
Then, I also carefully remove the larynx, but so not to mess up the neck roast on a buck or a large doe. I do Chicago neck tie cut under the jaw at the throat and sever the larynx and then reach up inside the cavity and pull it out of the neck. Takes a bit of effort, but then you don't have to expose the neck meat in the field. You don't want the larynx left in the deer. If I can't get it out in the field, then I'll remove it when I hang and skin the deer.
Then turn the carcass over on it's belly to drain while I clean up a bit.
Then get that critter hanging from the back legs with a gambrel and skin as soon as possible if butchering yourself (highly recommended). Skins much easer when the carcass is still warm and not stiff. Also, by skinning quickly the carcass cools much faster - better for the meat. Let hang at least a couple days or longer if not too warm. I have a commercial freezing unit that I operate in my garage during deer seasons if it